I need to take tricky integral without Fourier transform

Question

$$ \int_{0}^{inf} \dfrac{\vec{v} [\vec{n} \times \vec{n} \times \dfrac{d\vec{v}}{dt}]}{|\vec{r}-\vec{a}|} \delta(\vec{r}-\vec{a(t)}) \,dr $$ I took the inmtegral with Fourier transform, but teacher said to find way to do it without it. I tried to write explicitely cross-cross product in spherical coordinates but this gave no benefit (don't want to write with LaTeX 3 lines of useless from my point of view fromula). Is there any other elegant way to achieve result?